how to split an object into two
In Edit mode you need to select at least one face or one edge of your object, or the whole mesh, meaning you want to cut this face/edge/whole mesh. Then activate the Bisect tool. On the bottom of the Tools panel (on the left of the 3D View) you can enable some options. I guess the best is to leave everything as it is. Then to separate the two parts you've created, keep the cutting selected, then press V to rip, now you can select a half or the other with L. To make another object with one of your halves, select it and press P > Selection.